See-Saw With Hinge - Rigid Body Physics Simulation Not Working Properly
The Solution to the above and just about all catapult/see-saw type physics simulations lies in two important steps :


Properly Set up the physics :


a.) Set the falling object/cube as "Active" Rigid body.

b.) Set the Catapult (or see-saw object) as "Active" Rigid body. Do NOT Set this as "animated" unless you want this simulation to be controlled through the animation system or key frames.

c.) Set the Hinge as a "PASSIVE" Rigid body.


Set up Of the Catapult &amp; Hinge constraint : 


a.) Select The Hinge first, then shift select the catapult. This Will create an "empty object like Axis" which is the constraint. Set This constraint type to "Hinge". This can be selected on the Left Panels Menu by pressing "N" or under the "Constraint" Physics menu.

b). Set Up Proper Axis For The Hinge Constraint. An easy way to do this is to set all rotations of this constraint to zero. Then based on your object's orientation, select the proper axis for it to work. Trial &amp; error may be required but with the above suggestions it should take not take more than a minute or two.

-- Thanks to @m.ardito for always trying to help and provide hints. Thanks To @Rich Sedman for providing a useful explanation. Thanks To @Mr Zak &amp; @cegaton for providing help on question editor image tool.